Factorization

Factoring is the operation of rewriting polynomials into product form. Below are two examples, one easy and one difficult.

Exemplification 1

Example: x^2 - 4

Solution: This can be factored into (x-2)(x+2)using the square difference formula.

Exemplification 2

E: x^2 - 5x + 6

S: Solving using factorization yields (x-2)(x-3). Thus, x=2 or x=3.

Question

① (x-3)(x-1)(x+5)(x+7) - 960

S: First, expand and organize.

(x-3)(x-1) = x^2 - 4x + 3

(x+5)(x+7) = x^2 + 12x + 35

Next, these are multiplied together.

(x^2 - 4x + 3)(x^2 + 12x + 35)

If you expand this...x^4 + 8x^3 - 13x^2 - 132x + 105 になります。

Finally, subtract 960.

x^4 + 8x^3 - 13x^2 - 132x + 105 - 960 = x^4 + 8x^3 - 13x^2 - 132x - 855

The final factorization is as follows x^4 + 8x^3 - 13x^2 - 132x - 855 です。

② a^2 - b^2 - 4c^2 - 6a + 4bc + 9

S: Organize and factorize.

First, a^2 - 6a + 9 becomes (a - 3)^2.

Then b^2 - 4bc + 4c^2 is (b - 2c)^2.

Thus, factoring the whole, we get (a - 3)^2 - (b - 2c)^2.

Factor this using the square difference formula.

(a - 3 + b - 2c)(a - 3 - b + 2c)

③ (x^2 - 15x) + (x^2 - 225)

S: First, find the common factor.

x^2 - 15x + x^2 - 225 = 2x^2 - 15x - 225

This is further factorized.

2(x^2 - 7.5x - 112.5) = 2(x - 15)(x + 7.5)

④ x^4(2x - 3y) + 27(9y - 6x)

S:First, find common factors.

Factor out 27.

x^4(2x - 3y) - 27(6x - 9y) = x^4(2x - 3y) - 27(3(2x - 3y))

Factor out the common factor (2x - 3y).

(2x - 3y)(x^4 - 27)

Expansion of the formula

Expansion is the operation of rewriting a polynomial product in the form of a sum. Below are two examples of easy and difficult problems.

Example 1/h3>

E: (x + 2)(x + 3)

S: This can be expanded to x^2 + 5x + 6 using the distributive law.

Example 2

E: (2x - 3)(x + 4)

S: This can also be expanded to 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12 using the distributive law.

Question

① Expand (x-3)(x-1)(x+5)(x+7) - 960 and factor again.

S: (x-3)(x-1) = x^2 - 4x + 3

(x+5)(x+7) = x^2 + 12x + 35

This is multiplied and expanded.

(x^2 - 4x + 3)(x^2 + 12x + 35) = x^4 + 8x^3 - 13x^2 - 132x + 105

Finally, subtract 960 and re-factorize.

x^4 + 8x^3 - 13x^2 - 132x - 855

② Expand a^2 - b^2 - 4c^2 - 6a + 4bc + 9 and factor it similarly.

S: a^2 - 6a + 9 is (a-3)^2.

b^2 - 4bc + 4c^2 は (b-2c)^2 です。

a^2 - 6a + 9 is (a-3)^2.

(a-3)^2 - (b-2c)^2 = (a-3+b-2c)(a-3-b+2c)

③ Simplify by expanding (x^2 - 15x) + (x^2 - 225).

S: x^2 - 15x + x^2 - 225 = 2x^2 - 15x - 225

Factorize further.

2(x^2 - 7.5x - 112.5) = 2(x-15)(x+7.5)

④ x^4(2x - 3y) + 27(9y - 6x) to expand and organize.

S: First, find the common factor.

x^4(2x - 3y) - 27(6x - 9y)

factor out 27.

x^4(2x - 3y) - 27(3(2x - 3y))

Factor out the common factor (2x - 3y).

(2x - 3y)(x^4 - 27)